# Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^*$-algebras, and bivariant homology

### Snigdhayan Mahanta

University of Regensburg, Germany

## Abstract

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC_\infty^*}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC_\infty^*}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC_\infty^*}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connective $\mathtt E$-theory and compute the connective $\mathtt E$-theory groups of $\mathcal{O}_\infty$-stable $C^*$-algebras. We also introduce and study the nonconnective version of Quillen's nonunital $\mathtt K'$-theory in the framework of stable $\infty$-categories. This is done in order to promote our earlier result relating topological $\mathbb T$-duality to noncommutative motives to the $\infty$-categorical setup. Finally, we carry out some computations in the case of stable and $\mathcal{O}_\infty$-stable $C^*$-algebras.

## Cite this article

Snigdhayan Mahanta, Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^*$-algebras, and bivariant homology. J. Noncommut. Geom. 10 (2016), no. 4, pp. 1269–1301

DOI 10.4171/JNCG/260